# Binary Tree to Segment Tree

Important data structures used during computer science to address a range of issues include binary trees & segment trees. A segment tree is a specialised data structure in use for ranged searches and updates on an array, whereas a binary tree is still a tree-like data structure in which each node has no more than two children. We will examine how well a binary tree could be transformed into a segments tree in this article, as well as how this transformation may be helpful in resolving particular difficulties.

Let's first examine more closely at binary trees. A binary tree is made up of nodes with a maximum of two offspring, known as that of the left and right children. A value or even a pointer to some other data structure can be stored in each node's data storage. The root node is the topmost node in the tree, and using straightforward algorithms like depth-first search or breadth-first search, each node can be traversed iteratively.

Let us just think about segment trees now. For range searches and array updates, a segment tree is indeed a specialised data structure. A array of values is split into two halves recursively until one value is present in each segment, at which point the segment tree is constructed. The information about each node inside the segment tree, such as the segment's sum or minimum value, is a representation of a segment of a original array.

We must come up with a method to represent all segments of the initial array in the binary tree in order to transform it into a segment tree. One typical method is to link each node of the binary tree to a section of the initial array. The left and right offspring of each node indicate the left and right sides of the segment represented either by parent node, while the root node represents the complete array.

We can use the data contained in the binary tree nodes to record the details about each segment inside the segment tree. For instance, we may store the total of the values for each segment inside the binary tree node for that segment. Then, we can use this knowledge to swiftly respond to range queries and changes on the initial array.

Several situations call for the conversion of a binary tree into a segment tree. Solving issues involving accessing & updating ranges of array members is one common use. Consider the situation where you need to calculate the sum of all the entries between indexes l and r for an array of integers. We can respond to this question in O(log n) time to use a segment tree, where n represents the dimension of the array. Similar to this, we may update a value of just one element or a set of the array's components in O(log n) time using a segment tree.

Let's use the binary tree shown below as an illustration:

We must first assign depth-first indices to each node in order to transform this binary tree together into segment tree. The nodes will now be numbered as follows:

Each node has a number that follows the node format (index). Recursively assigning indices to a node's left and right children begins with the root node 1, which is given index 1, and proceeds from there. We provide each node two times its own index as its left child's index and two times its very own value plus one as its right child's index.

We can now generate the segment tree after giving each node an index. Each element of the array representing the segment tree will represent a section of the initial binary tree. The ith component of the array would specifically match the binary tree segment with index i.

Starting with an array with length 2n-1, where n represents the total amount of nodes inside the binary tree, we can construct the segment tree. If n=7 in our example binary tree, the segment tree will be 2n-1 length, or 13. We set the array's elements all to 0 at the beginning.

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

The segment tree's leaf nodes are then filled in. The leaf nodes match the binary tree's original nodes. We set that appropriate array component to the result of the leaf node for each leaf node.

[0, 4, 5, 6, 7, 0, 0, 0, 0, 0, 0, 0, 0]

We next compute the value of each individual element as that of the sum of it's own two children and iteratively construct the segment tree. For instance, the segment tree's node 2 will have a value equal to the sum of a values of the its children 4, and 5. This process is continued until the root node's value, which is the binary tree's total value, has been determined.

As a result, the transformation of a binary tree into a segment tree offers a strong tool for handling issues with range queries and array updates. We may efficiently perform searches and updates by connecting each branch inside the binary tree with such a segment of a array and storing the information about every segment in the associated node of a binary tree.